# Lec15Reduc - Bryan Pearsaul Jesse Phillips Discrete...

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Bryan Pearsaul Jesse Phillips Discrete Structures II Reducibility To reduce a problem A to a problem B is to show that if you had a solution to B then you can also solve A. Given: A TM = {<M, w> | M is a TM and it accepts w} A TM is undecidable. Prove: HALT TM = {<M, w> | M is a TM and halts on w} is also undecidable. Goal: If we can reduce A TM to HALT TM , then HALT TM is undecidable as well. 1. Assume a TM exists that decides HALT TM , call this R. 2. Produce S: a machine that decides A TM S has input (M, w) 1). Call R(M, w) 2). If R rejects, reject (because we know that w L(M)) else run M on w and return its answer Therefore we have shown A TM to be decidable, which we know is not true. So our initial assumption is false. Prove: E TM = {<M> | M is a TM and L(M) = Ø ) is undecidable. Goal: If we can reduce A TM to E TM , then E TM is undecidable as well. 1. Assume a TM exists that decides E TM , call this R. 2. Produce a machine to solve A TM (We’re stuck if we just pass <M> to R, so we must pass a different machine as a parameter) // we’re never going to run M , we will just use it as input to R. Create a new machine M from M and w whose language is L(M ) = { Ø , {w}}. 1). M

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Lec15Reduc - Bryan Pearsaul Jesse Phillips Discrete...

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